Let \(\mathcal {H}_{s+t}\)  be the \(\sigma \) -field by \((T_j)_{j:T_j\le s+t},N(s),N(s+t)\)  and \((L_j(t-T_j))_{j:T_j\le s+t}\) , so that \(\mathcal {G}_s\subset \mathcal {H}_{s+t}\) . Write
\(M(s,s+t] = \sum _{j=1}^{N(s)} L_j (s-T_j,s+t-T_j] +\sum _{j=N(s)+1}^{N(s+t)}L_j (s+t-T_j)\) 
and take conditional expectation on \(\mathcal {G}_s\) ,
\(& {\mathbb {E}}[\sum _{j=1}^{N(s)}L_j(s-T_j,s+t-T_j] \mid \mathcal {G}_s] + {\mathbb {E}}\,{\mathbb {E}}[\sum _{j=N(s)+1}^{N(s+t)} L_j(s+t-T_j)\mid \mathcal {H}_s] \mid \mathcal {G}_s ] \\&= \sum _{j=1}^{N(s)} \mu (s-T_j,s+t-T_j]+ {\mathbb {E}}[\sum _{j=N(s)+1}^{N(s+t)} \mu (s+t-T_j)],\) 

where we use the repeated expectation [1]} argument together with Theorem
REF . In the last expression, notice that the sequence \((T_j)_{N(s)+1\le j\le N(s+t)}\)  is symmetric and given \(N\)  and \(\eta \) , we could
regard it as an iid sequence such that \(T_j\sim \eta (dx)/\eta (s,s+t],\,s<x\le s+t\, a.s.\)  Hence a conditional
argument gives the first part of (REF ). The second
expression of (REF ) is obtained by differentiating LP
of the stochastic integral with \(\eta \)  (cf. [2]}).
For the expression (), we write
\((M(s,s+t])^2 &= \big ( \sum _{j=1}^{N(s)} L_j(s-T_j,s+t-T_j)\big )^2 + 2 \sum _{j=1}^{N(s)}L_j(s-T_j,s+t-T_j]\sum _{j=N(s)+1}^{N(s+t)}L_j(s+t-T_j) \\&+ \big (\sum _{j=N(s)+1}^{N(s+t)} L_j(s+t-T_j)\big )^2 =: \mathrm {I} + 2\mathrm {II} +\mathrm {III}\) 

and take conditional expectations for these quantities, where the
repeated expectation argument together with Theorem REF  are
again used. Then we obtain
\({\mathbb {E}}[\mathrm {I} \mid \mathcal {G}_s] &= \sum _{j=1}^{N(s)} (\mu ^2+\sigma ^2)(s-T_j,s+t-T_j] + \sum _{j\ne k,\,j,k=N(s)+1}^{N(s)} \mu (s-T_j,s+t-T_j]\mu (s-T_k,s+t-T_k], \\{\mathbb {E}}[\mathrm {II} \mid \mathcal {G}_s] &= \sum _{j=1}^{N(s)}\mu (s-T_j,s+t-T_j] {\mathbb {E}}[\sum _{j= N(s)+1}^{N(s+t)} \mu (s+t-T_j)], \\{\mathbb {E}}[\mathrm {III} \mid \mathcal {G}_s] &= {\mathbb {E}}[\sum _{j=N(s)+1}^{N(s+t)} (\mu ^2+\sigma ^2)(s+t-T_j)] + {\mathbb {E}}[\sum _{j\ne k,\,j,k=N(s)+1}^{N(s+t)} \mu (s+t-T_j) \mu (s+t-T_k)],\) 

so that
\(\mathrm {Var} (M(s,s+t] \mid \mathcal {G}_s) &= {\mathbb {E}}[\mathrm {I}+2\mathrm {II}+\mathrm {III} \mid \mathcal {G}_s]-\big (E [M(s,s+t] \mid \mathcal {G}_s]\big )^2 \\&= \sum _{j=1}^{N(s)}\sigma ^2 (s-T_j,s+t-T_j)+ {\mathbb {E}}[\sum _{j=N(s)+1}^{N(s+t)} (\mu ^2+\sigma ^2)(s+t-T_j)]\\&\quad + {\mathbb {E}}[\sum _{j\ne k,\,j,k=N(s)+1}^{N(s+t)}\mu (s+t-T_j)\mu (s+t-T_k)] -\big ( {\mathbb {E}}[\sum _{j=N(s)+1}^{N(s+t)}\mu (s+t-T_j) ]\big )^2.\) 

Now since the conditional order statistic property of \((T_j)\)  yields
\({\mathbb {E}}[\sum _{j=N(s)+1}^{N(s+t)}(\mu ^2+\sigma ^2)(s+t-T_j)] &= {\mathbb {E}}[\int _{(s,s+t]}(\mu ^2+\sigma ^2)(s+t-u)\eta (du)], \\{\mathbb {E}}[\sum _{j\ne k,\,j,k=N(s)+1}^{N(s+t)} \mu (s+t-T_j)] &={\mathbb {E}}[ \big ( \int _{(s,s+t]}\mu (s+t-u)\eta (du) \big )^2],\) 

we obtain (). The second expression is obtained again
by \(LP\)  of the stochastic integrals with \(\eta \) .
